Plotting Integers

We visualize a number's location within the real number set by plotting its coordinate on the real number line. The number line is filled with tick marks, each marking the coordinate of a different number on the line. To the plot the location of a number on the line, we place a heavy dot over its coordinate on the line.

Ex.1: Plot the number 3 on the real number line

Ex.2: Plot the number 10 on the real number line

Plotting Negative Numbers

As we move to the right on the number line, we see that the value of the numbers will continue to increase the further we travel. But, as we travel to the left, the value of the numbers decrease. If we travel left past the origin, we will start to see negative numbers (or numbers that are less than zero). The further we travel, the more negative the numbers will become. To plot a negative number, we place a heavy dot on the line over the number's coordinate.

For example, to plot the number -3, we travel left past the origin and place a dot over the number -3.

Ex.3: Plot the number -3 on the number line

Ex.4: Plot the number -10 on the number line

Plotting Fractions

Fractions are numbers that are between integers. To plot a fraction on the real number line, it is typically best to first convert the value into a mixed number or decimal. This will allow us to easily see its integer and fractional parts. To plot the number, we first move right by the number of units equal to the integer part of the number (move left instead of right if the number is negative). Then we move further a distance proportional to the fractional part of the number. The final location is the coordinate.

For example, to plot the number 3¹⁄₂ on the real number line, we first move right 3 units. Once there we move forward ¹⁄₂ unit (or halfway between the ticks 3 and 4), then plot the coordinate.

Ex.5: Plot the number 3¹⁄₂ on the number line.

To plot a negative fraction, we move left of the origin instead of right. First to the integer part of the number, then further left an amount proportional to the fraction.

For example, to plot the point -0.75, we first move left the number of units equal to the integer part of the number. Then, we move left the fractional part of the number, which is 0.75, or ³⁄₄ of the way to -1. This is easier if we place fractional tick marks.

Ex.6: Plot the number -0.75 on the number line.

Plotting Sets

To plot a set on the real number line, we simply plot each number in the set on the line one-by-one.

Ex.7: Plot the set A={1,3,8,10} on the real number line.

Ex.8: Plot the set B={-4,-1,0,2,5} on the real number line.

Scaling the Number Line

We scale the number line to: make sure it has room for all the numbers we're adding; and, to illustrate the numbers on it as effectively as possible.

For example, to illustrate the set A={0,25,50,100} we need to shrink the line so that all the numbers from 0 to 100 are visible. That means squeezing the tick marks together and increasing the intervals between them.

Ex.9: Plot the set A={0,25,50,100} on the real number line.

Sometimes we need to expand the number line to better view the plots of smaller numbers.

For example, to graph the set B={0, ¹⁄₄, ¹⁄₂, ³⁄₄, ⁷⁄₈}, we need to expand the number line so that we can see the individual numbers.

Ex.10: Plot the set B={0, ¹⁄₄, ¹⁄₂, ³⁄₄, ⁷⁄₈} on the real number line.

Comparing and Ordering Values

To compare multiple values using the number line, we plot each coordinate on the line and then compare them to each other. If a coordinate is to the right of another, then that is the coordinate of the larger number. Between this logic and the graph we can determine the order of values.

Ex.11: Fill in the box with <, >, or = to make the statement true.

Answer: 1 < 8

Using the number line to order values becomes more handy when the values are more obscure, such as with fractions.

Ex.12: Order the set A={¹⁄₃, ¹⁄₂, ⁷⁄₈, ⁵⁄₈, ¹³⁄₁₆}.

Answer: A={¹⁄₃, ¹⁄₂, ⁵⁄₈, ¹³⁄₁₆, ⁷⁄₈}

Adding on the Real Number Line

Adding numbers with the real number line allows us to illustrate the process of addition and visualize the sum in comparison to its addends. To add on the real number line, we simply move to the coordinate of the first addend, then we move to the right by the number of units in the second addend (if the 2nd addend is negative we move to the left by the number of units instead).

For example, if we want to add -5+8 using the real number line, first move to the coordinate of -5 (which is to the left of the origin). Then, move to the right 8 units.

Ex.13: Add -5+8 using the real number line

The number line can also help us add fractions.

For example, to add 3¹⁄₂ + 2¹⁄₂, first move to the coordinate of 3¹⁄₂. Then, move to the right 2¹⁄₂ units.

Ex.14: Add 3¹⁄₂ + 2¹⁄₂ using the real number line

Subtracting on the Real Number Line

As with addition, subtracting with the real number line allows us to illustrate the process of subtraction and visualize the difference in comparison to the minuend and subtrahend. There are 2 methods that we can use to subtract with the number line, either we can Count-Back or Count-Up. To Count-Back, we start by moving to the coordinate of the minuend on the number line. Then, we count-back (move to the left) by the number of units in the subtrahend to arrive at the difference.

For example, to perform the subtraction 9-5 using the Count-Back method, we start by moving to the coordinate of the minuend 9 on the number line. Then, we count-back 5 units.

Ex.15: Perform the subtraction 9 - 5 using the Count-Back method on the real number line.

We can also use our second method of subtraction, Counting-Up, to solve the same example. To Count-Up, we start by moving to the coordinate of the subtrahend (5); then, we move right, counting-up the number line until we reach the coordinate of the minuend (9). The difference is the number of units it takes to reach the minuend from the subtrahend.

Ex.16: Perform the subtraction 9 - 5 using the Count-Up method on the real number line.

Multiplication on the Real Number Line

We can use the real number line to illustrate the process of multiplication, and find the product. Multiplying on the number line is similar to multiplying in real life, in that we add the same quantity multiple times. However, instead of counting in groups, we count in jumps up the number line by the number of units equal to the multiplier.

For example, to multiply 7 × 5 on the number line, we jump up the line 7 times by 5 units a jump to arrive at the final product.

Ex.17: Multiply 7 × 5 using the real number line.

If one of the numbers we are multiply by is negative, then we jump to the left instead of right.

For example, to multiply 7 × -2 we jump 7 times jumping left 2 units each jump to get to the product.

Ex.18: Multiply 7 × -2 using the real number line.

Division on the Real Number Line

As with multiplication, we can use the real number line to illustrate the process of division. We do this by finding the number of times the divisor fits into the dividend. To perform division on the real number line first thing we do is plot the dividend on the number line. Then, starting at the origin, we move toward the dividend jumping right by the number of units in the divisor. When we reach the dividend, the number of jumps it took is the quotient.

For example, to find the quotient 16 ÷ 4, we start by plotting 16 (the dividend) on the number line. Then, starting at the origin, we move toward the dividend jumping 4 units a time. The quotient will be the number of jumps it takes to reach the dividend.

Ex.19: Divide 16 ÷ 4 using the real number line.

Sometimes when dividing on the number line, we jump over the dividend before we land on it. This means that the quotient has a remainder (or a fractional part). To find the remainder, go back to the number before going over the dividend. Then, count the number of units between that number and the dividend. The difference will be the remainder. We then take the remainder and place it over the divisor in a fraction. The quotient will be the number of jumps plus the fraction, with the remainder and divisor.

For example, to find the quotient of 25 ÷ 6, we start by plotting 25. Then, starting at the origin, we jump toward the dividend 6 units at a time. However, on the 5th jump, we jump over the dividend and land on the number 30. So we go back, and count the number of units between the last jump and the dividend (which is 1 unit). The quotient is the number of jumps we made (4), plus fraction of the remainder (1) divided by the divisor (6).

Ex.20: Divide 25 ÷ 6 using the real number line.